Can I edit the motion path?
No, you can not interact with the Motion Path (the white dotted line with yellow keyframe indicators). It is merely a visual reference.

You can edit the motion of the object ...


by setting additional keyframes.
by changing the keyframe interpolation or bezier types in the dopesheet 
by changing the F-Curves of the animated channels in the graph editor
by animating it along a path with a Follow Path Constraint




Editability of the motion path may be achieved through the addon Motion Trail Addon:


Blender Addon List
BlenderArtist post



Motion Path Addon (gif)